## Discrete Cylindrical Vector Beam Generation from an Array of Optical Fibers

Optics Express, Vol. 17, Issue 16, pp. 13982-13988 (2009)

http://dx.doi.org/10.1364/OE.17.013982

Acrobat PDF (582 KB)

### Abstract

A novel method is presented for the beam shaping of far field intensity distributions of coherently combined fiber arrays. The fibers are arranged uniformly on the perimeter of a circle, and the linearly polarized beams of equal shape are superimposed such that the far field pattern represents an effective radially polarized vector beam, or discrete cylindrical vector (DCV) beam. The DCV beam is produced by three or more beams that each individually have a varying polarization vector. The beams are appropriately distributed in the near field such that the far field intensity distribution has a central null. This result is in contrast to the situation of parallel linearly polarized beams, where the intensity peaks on axis.

© 2009 Optical Society of America

## 1. Introduction

1. T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express **14**(25), 12188–12195 (2006). [CrossRef] [PubMed]

5. T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-Synchronous and Self-Referenced Coherent Beam Combination for Large Optical Arrays,” IEEE J. Sel. Top. Quant. El. **13**(3), 480–486 (2007). [CrossRef]

6. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. **77**(21), 3322 (2000). [CrossRef]

6. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. **77**(21), 3322 (2000). [CrossRef]

19. I. Moshe, S. Jackel, A. Meir, Y. Lumer, and E. Leibush, “2 kW, M2 ¡ 10 radially polarized beams from aberration-compensated rod-based Nd:YAG lasers,” Opt. Lett. **32**(1), 47–49 (2007). [CrossRef]

23. M. Rioux, R. Tremblay, and P.-A. Belanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. **17**(10), 1532 (1978). [CrossRef] [PubMed]

24. Y. I. Salamin, “Mono-energetic GeV electrons from ionization in a radially polarized laser beam,” Opt. Lett. **32**(1), 90–92 (2007). [CrossRef]

25. S. M. Iftiquar and J. Opt , “A tunable doughnut laser beam for cold-atom experiments,” B: Quantum Semiclass. Opt. **5**(1), 40–43 (2003). [CrossRef]

26. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express **7**(2), 77–87 (2000). [CrossRef] [PubMed]

27. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. **91**(23), 233901 (2003). [CrossRef] [PubMed]

28. T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. **203**(1–2), 1–5 (2002). [CrossRef]

31. D. J. Armstrong, M. C. Phillips, and A. V. Smith, “Generation of radially polarized beams with an image-rotating resonator,” Appl. Opt. **42**(18), 3550–3554 (2003). [CrossRef] [PubMed]

8. T. Moser, J. Balmer, D. Delbeke, P. Muys, S. Verstuyft, and R. Baets, “Intracavity generation of radially polarized CO2 laser beams based on a simple binary dielectric diffraction grating,” Appl. Opt. **45**(33), 8517–8522 (2006). [CrossRef] [PubMed]

11. A. V. Nesterov and V. G. Niziev, J. Phys. D Appl. Phys. **33**(15), 1817–1822 (2000). [CrossRef]

7. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. **32**(11), 1468–1470 (2007). [CrossRef] [PubMed]

19. I. Moshe, S. Jackel, A. Meir, Y. Lumer, and E. Leibush, “2 kW, M2 ¡ 10 radially polarized beams from aberration-compensated rod-based Nd:YAG lasers,” Opt. Lett. **32**(1), 47–49 (2007). [CrossRef]

## 2. Method and Experimental Setup

**E**(

*ρ*,

*z*) -

*k*

^{2}

_{0}

**E**(

*ρ*,

*z*) = 0. We use cylindrical coordinates, so that

*ρ*is the usual transverse coordinate,

*k*

_{0}=

*ω*/

*c*, and the usual sinusoidal time dependence has been factored out. Applying Lagrange’s formula permits the wave equation to be written in terms of the vector Laplacian: ∇

^{2}

**E**(

*ρ*,

*z*) +

*k*

^{2}

_{0}

**E**(

*ρ*,

*z*) = 0, which can then be reduced to the paraxial wave equation. The paraxial solutions are then inserted into the familiar Fraunhofer diffraction integral, which for propagation at sufficiently large

*z*, yields the electric field,

**E**

^{0}(

**r**+

**s**

_{j},0) is the incident electric field in the plane of the fiber array. It is clear that longitudinal polarization is not considered here, consistent with the paraxial approximation. The far field intensity will be a complex diffraction pattern that depends on the individual beam intensity profile in the near field as well its polarization state and optical phase. A diagram illustrating the geometry used is shown in Fig. 1, where an array of three circular holes of radius

*a*are equally distributed on the circumference of a circle of radius

*R*. The center of each hole is located at

**s**

_{j}=

*R*

**r**̂

_{j}, where

**r**̂

_{j}= (

**x**̂ cos

*θ*+

_{j}**y**̂ sin

*θ*) is the unit radial vector, and

_{j}**r**is the coordinate relative to each center. The center of each fiber is separated by an angle

*θ*= 2

_{j}*π*(

*j*- 1)/

*N*, where

*N*is the number of beams. One can scale to any number of beams, constrained mainly by the radius

*R*. For identical Gaussian beams, the incident field is expressed as,

**E**

^{0}=

*E*

^{0}(

**r**)

**r**̂

_{j}, where

*E*

^{0}(

**r**) =

*E*

_{0}

*e*

^{-r2/w20}. This permits Eq. (1) to be separated,

*E*

_{0}

*k*

_{0}/

*z*∫

^{a}

_{0}

*rdr*

*J*

_{0}(

*k*

_{0}

*rρ*/

*z*)e

^{-r2/w20}, and the prefactors that do not contribute to the intensity have been suppressed. It is clear from Eq. (2) that the far field transform preserves the radially symmetric polarization state of the system, as expected. Note that in the limit

*w*

_{0}/

*a*≫ 1, and using the relation

*xJ*

_{0}(

*x*) = [

*x*

*J*

_{1}(

*x*)]′, we have ℱ(

*ρ*,

*z*) ≈

*E*

_{0}(

*a*/

*ρ*)

*J*

_{1}(

*k*

_{0}

*aρ*/

*z*), which gives the expected Fraunhofer diffraction pattern for a circular aperture of radius

*a*. In the opposite limit, where the Gaussian profile is narrow enough so that the aperture geometry has little effect, we have ℱ(

*ρ*,

*z*) ≈

*E*

_{0}

*k*

_{0}

*w*

^{2}

_{0}(2

*z*) exp[-(

*k*

_{0}

*ρ*

*w*

_{0}/(2

*z*))

^{2}]. For the multiple beam arrangements investigated, corresponding to

*N*= 3,4,6, we then can calculate the intensity,

*I*(

_{N}*ρ*,

*ϕ*), explicitly:

*I*= ℓ

_{N}^{2}(

*ρ*,

*z*)퓚

_{N}(

*ϕ*;

*ρ*,

*z*),

*k*= (

_{x}*k*

_{0}/

*z*)

*ρ*cos

*ϕ*and

*k*= (

_{y}*k*

_{0}/

*z*)

*ρ*sin

*ϕ*. To rotate the array, one can perform a standard rotation

**r**′ = (

*ϕ*′)

**r**, so e.g., a

*π*/4 rotation would give, 𝓚

_{4}(

*ϕ*′) = 4(1 - cos(√2

*k*) cos(√2

_{x}R*k*)).

_{y}R*λ*= 1.064

*μ*m.

*N*= 4) involves expanding a beam from the Nd:YAG by means of a telescope beam expander to a diameter of about 1cm. The diffractive optical element then creates an 8 × 8 array of beams. Four of the beams are reflected and made nearly parallel by a segmented mirror, and then linearly polarized upon passing through a set of four half-wave plates. The phases of each beam are made identical by having them pass through articulated glass slides.

*N*= 3 and 6), which is simpler for larger arrays, is shown in Fig. 2. In this method the Nd:YAG is coupled into a polarization maintaining fiber. This signal is then split into multiple copies by means of a lithium niobate waveguide 8-way splitter and 8-channel electro-optic modulator. Each path can then have a separate phase modulation to ensure proper phasing of the beams. These signals are then propagated through fiber and coupled out and collimated by a 1 in. lens. The lens is slightly overfilled so the Gaussian outputs of the fibers are truncated at 1.1 × (the

*e*

^{-1}radius).

## 3. Discussion

*N*= 4 (middle row), the central null is surrounded by a checkerboard peak structure. This pattern clearly differs from the linear polarization by a rotation of

*π*/4, based on a simple phase argument. The

*N*= 6 case is shown in the bottom row, where the intensity vanishes at the origin, followed by the formation of bright hexagonal rings. The 3 beam configuration is shown in the top row, where again there is satisfactory agreement between the calculated and measured results. Any observed discrepancies may be reduced with an appropriately incorporated feedback system.

*ρ*, and normalize it to the intensity integrated over the entire image plane. This fraction, 𝒰

_{N}, can be determined for a given number of fibers,

*N*. If one considers a circular region that extends up to the primary peaked intensity patterns, it is often possible to accurately calculate this analytically. This is shown in particular for the

*N*= 4 case, 𝒰

_{4}, where,

*w*

_{0}= 42

*μ*m and

*R*= 2.7

*w*

_{0}into Eq. (6), we find 𝒰

_{4}≈ 0.57, which effectively characterizes the dominant field distribution contained within the given circle. Performing an analogous calculation at the same radius for

*N*= 6,8 and 10 yields 𝒰

_{6}≈ 0.69, 𝒰

_{8}≈ 0.81, and 𝒰

_{10}≈ 0.84, respectively. Thus, as expected, increasing the number of combined beams generally enhances the fraction, 𝒰

_{N}, and ultimately the desired radially polarized DCV beam emerges.

*λ*/2 retardation plate in different orientations in conjunction with a polarizing beam splitter.

5. T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-Synchronous and Self-Referenced Coherent Beam Combination for Large Optical Arrays,” IEEE J. Sel. Top. Quant. El. **13**(3), 480–486 (2007). [CrossRef]

## Acknowledgments

## References and links

1. | T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express |

2. | H. Bruesselbach, D. C. Jones, M. S. Mangir, M. I. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. |

3. | T. B. Simpson, F. Doft, P. R. Peterson, and A. Gavrielides, “Coherent combining of spectrally broadened fiber lasers,” Opt. Express |

4. | A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express |

5. | T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-Synchronous and Self-Referenced Coherent Beam Combination for Large Optical Arrays,” IEEE J. Sel. Top. Quant. El. |

6. | R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. |

7. | G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. |

8. | T. Moser, J. Balmer, D. Delbeke, P. Muys, S. Verstuyft, and R. Baets, “Intracavity generation of radially polarized CO2 laser beams based on a simple binary dielectric diffraction grating,” Appl. Opt. |

9. | N. Passilly, R. de Saint Denis, K. At-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A |

10. | S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. |

11. | A. V. Nesterov and V. G. Niziev, J. Phys. D Appl. Phys. |

12. | T. Hirayama, Y. Kozawa, T. Nakamura, and S. Sato, “Generation of a cylindrically symmetric, polarized laser beam with narrow linewidth and fine tunability,” Opt. Express |

13. | Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. |

14. | K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. |

15. | T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. |

16. | J. J. Wynne, “Generation of the rotationally symmetric TE01 and TM01 from a wavelength-tunable laser,” IEEE J. Quantum Electron. |

17. | Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. |

18. | Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. |

19. | I. Moshe, S. Jackel, A. Meir, Y. Lumer, and E. Leibush, “2 kW, M2 ¡ 10 radially polarized beams from aberration-compensated rod-based Nd:YAG lasers,” Opt. Lett. |

20. | I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. |

21. | M. Roth, E. Wyss, H. Glur, and H. P. Weber, “Generation of radially polarized beams in a Nd:YAG laser with self-adaptive overcompensation of the thermal lens,” Opt. Lett. |

22. | V. G. Niziev and A. V. Nestorov, and J. Phys, “D,” Appl. Phys. (Berl.) |

23. | M. Rioux, R. Tremblay, and P.-A. Belanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. |

24. | Y. I. Salamin, “Mono-energetic GeV electrons from ionization in a radially polarized laser beam,” Opt. Lett. |

25. | S. M. Iftiquar and J. Opt , “A tunable doughnut laser beam for cold-atom experiments,” B: Quantum Semiclass. Opt. |

26. | K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express |

27. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. |

28. | T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. |

29. | G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre Gaussian beams,” Opt. Commun. |

30. | J. L. Li, K. Ueda, M. Musha, A. Shirakawa, and L. X. Zhong, “Generation of radially polarized mode in Yb fiber laser by using a dual conical prism,” Opt. Lett. |

31. | D. J. Armstrong, M. C. Phillips, and A. V. Smith, “Generation of radially polarized beams with an image-rotating resonator,” Appl. Opt. |

**OCIS Codes**

(060.2310) Fiber optics and optical communications : Fiber optics

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(230.5440) Optical devices : Polarization-selective devices

(060.3510) Fiber optics and optical communications : Lasers, fiber

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: June 8, 2009

Revised Manuscript: July 14, 2009

Manuscript Accepted: July 17, 2009

Published: July 28, 2009

**Citation**

R. S. Kurti, Klaus Halterman, Ramesh K. Shori, and Michael J. Wardlaw, "Discrete cylindrical vector beam
generation from an array of optical
fibers," Opt. Express **17**, 13982-13988 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13982

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### References

- T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14(25), 12188–12195 (2006). [CrossRef] [PubMed]
- H. Bruesselbach, D. C. Jones, M. S. Mangir, M. I. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30(11), 1339 (2005). [CrossRef] [PubMed]
- T. B. Simpson, F. Doft, P. R. Peterson, and A. Gavrielides, “Coherent combining of spectrally broadened fiber lasers,” Opt. Express 15(18), 11731–11740 (2007). [CrossRef] [PubMed]
- A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express 10(21), 1167–1172 (2002). [PubMed]
- T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-Synchronous and Self-Referenced Coherent Beam Combination for Large Optical Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 480–486 (2007). [CrossRef]
- R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322 (2000). [CrossRef]
- G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32(11), 1468–1470 (2007). [CrossRef] [PubMed]
- T. Moser, J. Balmer, D. Delbeke, P. Muys, S. Verstuyft, and R. Baets, “Intracavity generation of radially polarized CO2 laser beams based on a simple binary dielectric diffraction grating,” Appl. Opt. 45(33), 8517–8522 (2006). [CrossRef] [PubMed]
- N. Passilly, R. de Saint Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22(5), 984 (2005). [CrossRef]
- S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234 (1990). [CrossRef] [PubMed]
- A. V. Nesterov and V. G. Niziev, J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000). [CrossRef]
- T. Hirayama, Y. Kozawa, T. Nakamura, and S. Sato, “Generation of a cylindrically symmetric, polarized laser beam with narrow linewidth and fine tunability,” Opt. Express 14(26), 12839–12845 (2006). [CrossRef] [PubMed]
- Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27(5), 285–287 (2002). [CrossRef]
- K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31(14), 2151–2153 (2006). [CrossRef] [PubMed]
- T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252(1-3), 12–21 (2005). [CrossRef]
- J. J. Wynne, “Generation of the rotationally symmetric TE01 and TM01 from a wavelength-tunable laser,” IEEE J. Quantum Electron. 10(2), 125–127 (1974). [CrossRef]
- Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30(22), 3063–3065 (2005). [CrossRef] [PubMed]
- Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. 31(17), 2619–2621 (2006). [CrossRef] [PubMed]
- I. Moshe, S. Jackel, A. Meir, Y. Lumer, and E. Leibush, “2 kW, M2 < 10 radially polarized beams from aberration-compensated rod-based Nd:YAG lasers,” Opt. Lett. 32(1), 47–49 (2007). [CrossRef]
- I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28(10), 807–809 (2003). [CrossRef] [PubMed]
- M. Roth, E. Wyss, H. Glur, and H. P. Weber, “Generation of radially polarized beams in a Nd:YAG laser with self-adaptive overcompensation of the thermal lens,” Opt. Lett. 30(13), 1665 (2005). [CrossRef] [PubMed]
- V. G. Niziev, A. V. Nestorov, and J. Phys D Appl. Phys. (Berl.) 32, 1455 (1999).
- M. Rioux, R. Tremblay, and P.-A. Belanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17(10), 1532 (1978). [CrossRef] [PubMed]
- Y. I. Salamin, “Mono-energetic GeV electrons from ionization in a radially polarized laser beam,” Opt. Lett. 32(1), 90–92 (2007). [CrossRef]
- S. M. Iftiquar and J. Opt, “A tunable doughnut laser beam for cold-atom experiments,” B: Quantum Semiclass. Opt. 5(1), 40–43 (2003). [CrossRef]
- K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]
- R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]
- T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203(1-2), 1–5 (2002). [CrossRef]
- G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre?Gaussian beams,” Opt. Commun. 237(1-3), 89–95 (2004). [CrossRef]
- J. L. Li, K. Ueda, M. Musha, A. Shirakawa, and L. X. Zhong, “Generation of radially polarized mode in Yb fiber laser by using a dual conical prism,” Opt. Lett. 31(20), 2969–2971 (2006). [CrossRef] [PubMed]
- D. J. Armstrong, M. C. Phillips, and A. V. Smith, “Generation of radially polarized beams with an image-rotating resonator,” Appl. Opt. 42(18), 3550–3554 (2003). [CrossRef] [PubMed]

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